Photobucket waited many years of allowing people to post their pics on Photobucket for free. they made it easy, and now? They have blocked everyone's pictures unless you pay a monthly access fee. Even spotify only charges $50 for an entire year. I'd pay one months subscription, and then copy all the pics, even if I have to screen-shot them.

make shared album (set for public access)
put photos into shared album
go to the shared album
go to the individual photo view in the shared album
copy url from this individual view of the photo in the shared album
paste it into the forum or website
test with non-logged in browser to see what others see and make sure it is working

It works when it is done properly, but it is easy to mess up. If you paste the link from the wrong window, or a non shared album it won't work for others, but will work for you when logged in.